From an arithmetical standpoint, the question introduces so many unknown variables that even a modern computer cannot handle the calculation. As part of a Technology Foundation STW project, Leentvaar split the problem into a number of less complex partial problems. He also developed an algorithm that, combined with computer technology, can calculate the option price for a basket of shares.
Using a computer, it is hard to determine the option price for a variety of shares. The multitude of possibilities mean the number of unknowns to be resolved grows exponentially. For example, an option on a basket of five shares involves 32 million unknowns, given a 32-point grid. This is beyond the capabilities of today’s computer systems. Leentvaar used the so-called thin-grid method to split the problem into a number of less complex partial problems that could be handled by a modern computer system. The option price can be estimated accurately by combining the solutions of all the partial problems in the correct manner.
However, the option contract has one annoying feature if the thin-grid method is used, namely it is not always advantageous to exercise an option (i.e. to exercise the right to buy or sell at a particular time). Mathematically, this leads to a ‘kink’ in the final solution to the problem. Leentvaar used variable transforms to minimise this ‘kink’, so that it is dependent on only one variable: the value of the basket itself. The thin-grid method can be used with reasonable accuracy by calculating this particular variable more precisely and the other variables more coarsely.
This leaves the options based on the worst- or best-performing share. These options do not lend themselves to solution using differential equations because the preconditions are missing. Leentvaar used advanced parallelisation methods (Fourier transforms) for this purpose. By cleverly splitting up the problem, these methods are able to solve each part independently of the rest. In this way, the researcher managed to combine the power of the thin-grid method with the parallelisation of the Fourier transforms into a computer model that divides a large problem into many small parts and then solves these.
Trade in underlying futures, or hedging, is based on derivatives of the option prices. This either cannot be done accurately enough using current methods or there is no reference. Leentvaar’s method is one where the derivatives, or Greeks, can easily be determined on the basis of the calculated prices. This offers the prospect of further research into making pricing methods more efficient. Participating organisations are ABN-AMRO, Rabobank, Binck (formerly AOT) and Tilburg University.
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